Listening to this, I was struck that Dave Swarbick actually sings: 'Last night I saw the old moon clear/with the new moon in her hair.' It struck me because the more usual lines are:
'I saw the new moon late yestreen
Wi' the auld moon in her arm;
And if we gang to sea, master,
I fear we'll come to harm.'
This is what Coleridge quotes at the beginning of 'Dejection, an Ode':
Late, late yestreen I saw the new Moon
With the old Moon in her arms;
And I fear, I fear my Master dear!
We shall have a deadly storm.
That's Coleridge, I think, quoting from memory and 'improving' upon the original; for although there are various versions of the original poem there aren't any, I think, quite a leaden as this. But the question is: were Fairport quoting some alternate original version, or did they just make it up?
But 'the new moon in the old moon's arms': Wikipedia has an entry on the phrase that takes us to the earthshine. It doesn't make sense, though: a new moon is (to quote the Great Infallible again) 'when the Moon is not visible to the naked eye.' To speak of seeing a new moon would be like speaking of seeing an invisible man. But why 'in her hair'? In what sense? Possible meanings: the old English for a February moon is 'wolf moon'. 'Hair' (as the OED points out) is used astronomically of the rays of the sun, of comets' tails etc ('yet shall the aged sun shed forth his hair', Marlowe and Nashe's Dido, 1594). This latter makes more sense to me: moonshine, in the sense of lunar crepuscular rays, are a function of atmospheric interference in observation, and more likely to happen when the air is disturbed, as before a storm. The same cannot be said of earthlight.